Detection of Signals With Observations in Multiple Subbands: A ...

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014

Detection of Signals With Observations in Multiple Subbands: A Scheme of Wideband Spectrum Sensing for Cognitive Radio With Multiple Antennas Taehun An, Student Member, IEEE, Iickho Song, Fellow, IEEE, Seungwon Lee, Student Member, IEEE, and Hwang-Ki Min

Abstract—We address detection schemes of spectrum sensing for cognitive radio with multiple receive antennas operating over a wideband channel composed of a multitude of subbands. By taking the observations in all subbands into consideration in the likelihood functions for sensing a subband, the test statistics of the proposed schemes are functions of the sample covariance matrix in the subband under consideration and that in the subband exhibiting the lowest power spectral density. The false alarm and detection probabilities of the proposed schemes are analyzed theoretically and confirmed via simulations when the numbers of observations are the same for all the subbands. It is shown through computer simulations that the proposed schemes can provide considerable performance gains over conventional schemes for wideband spectrum sensing when the observations are spatially correlated and temporally independent/dependent. Index Terms—Cognitive radio (CR), likelihood function, multiple receive antennas, noncoherent scheme, signal detection, wideband spectrum sensing (WSS).

I. I NTRODUCTION

I

N cognitive radio (CR) networks, secondary users are allowed to opportunistically use some portions of frequency bands licensed to others if no harmful interference is caused to the primary users [1]. Consequently, the function of spectrum sensing, determining the existence of a spectrum user reliably and promptly, constitutes one of the principal components in the CR systems [2]–[4]. The energy detector (ED) [5], a noncoherent scheme, is widely employed because of its simple implementation and reliable performance in Gaussian environment, but its performance degrades severely if there exists uncertainty in the noise variance [6]. To reduce the noise uncertainty, the CR is required to estimate the noise variance when employing the ED scheme [7].

Manuscript received December 13, 2013; revised March 16, 2014 and June 18, 2014; accepted August 5, 2014. Date of publication August 20, 2014; date of current version December 8, 2014. This work was supported by the National Research Foundation of Korea under Grant 2010-0015175 and by the Convergence Information Technology Research Center Program of the National IT Industry Promotion Agency under Grant NIPA-2014-H0401-14-1009, both with funding from the Ministry of Science, Information and Communications Technology, and Future Planning. The associate editor coordinating the review of this paper and approving it for publication was E. A. Jorswieck. The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TWC.2014.2349938

To achieve the desired detection performance under the uncertainty of noise variance, CR with multiple receive antennas has been explored in [8]–[13], where most spectrum sensing schemes determine the presence or absence of the signal of the spectrum user (SSU) based on the sample covariance matrix of the observation vector. The schemes of the arithmetic to geometric mean (AGM) [10] and maximum eigenvalue to sum of eigenvalues (MES) [11] have been derived by employing the generalized likelihood ratio test (GLRT) under the assumption that the channel gain and noise variance are unknown to the CR and the observations are time-wise independent. When the observations are temporally dependent, the schemes of the covariance absolute value (CAV) [8] and maximum to minimum eigenvalue (MME) [9] have been proposed based on the sample covariance matrix of consecutive observation vectors and shown to perform better than the other schemes designed for temporally independent observations. In the meantime, for CR systems operating over a multiple of frequency bands, wideband spectrum sensing (WSS) schemes searching over a multitude of frequency bands simultaneously have recently been considered in [14]–[21]. In WSS also, employing the ED scheme would be effective if the noise variance is known exactly. In the presence of correlation between the occupancies of frequency subbands, the multiband joint detection scheme [19] has been proposed based on the ED for cooperative spectrum sensing under the assumption that the noise variances and channel gains are known to the CR. On the other hand, under the uncertainty of noise variance, the detection scheme with the information on the minimum number of vacant subbands (DIM) has been proposed for the CR with a single receive antenna and shown to provide better detection performance than the ED scheme [16]. When the observations are temporally dependent, the feature detection scheme in [18] exploiting known statistical properties of the SSU has been proposed based on the GLRT. Although the problem of designing effective noncoherent detection schemes for WSS has been addressed in several studies, most studies (e.g., [14]–[17] and [19]) have considered the detection scheme for temporally independent observations. In this paper, we propose novel noncoherent detection schemes of WSS in spatially correlated, temporally independent/ dependent scenarios when the CR with multiple receive antennas operates over a wideband channel. Without assuming that the channel gains and variances of the SSU and noise are known

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AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

to the CR, the proposed schemes are derived from the likelihood functions which employ estimates of unknown parameters. In particular, based on the fact that with high probability at least one subband is vacant, which motivates the research of spectrum sharing or CR, and the intuition that the subband of observations with a lower power spectral density (PSD) is more likely to contain only noise than that with a higher PSD, the proposed schemes secure the information on the noise PSD from the observations in the subband with the lowest PSD. By appropriately taking the observations in all subbands into account for sensing a subband, the proposed schemes perform better than other schemes which sense a subband based on the observations only in the subband. This paper is clearly distinct from [5], [8]–[13], [15], and [17] in that the schemes proposed in this paper exploit observations in all the subbands for sensing a subband. Although the schemes in [14], [16], [20], and [21] similarly exploit observations in all the subbands for sensing a subband, this paper is distinguishable from [14], [16], [20], and [21] in the following points. First, we address detection schemes of WSS for temporally independent and temporally dependent observations. Second, the schemes proposed in this paper can be employed for WSS even when the noise variances are different from one subband to another. Third, the detection performances of the proposed schemes, which can be calculated numerically, are derived and confirmed via simulations.

II. S YSTEM M ODEL Consider a CR network system operating over a wideband channel divided into K non-overlapping subbands of center K frequencies {fk }K k=1 and bandwidths {Bk }k=1 . It is assumed that the CR is equipped with A receive antennas and that K {fk }K k=1 and {Bk }k=1 are known to the CR. When the wideband signal in the continuous-time domain is received at an antenna of the CR, it is passed through a bank of K bandpass filters, down converted to the baseband, and sampled. Then, the n-th observation, for n = 1, 2, . . . , Nk , can be expressed as (q)

(q)

(q)

Rk (n) = Hk Sk (n) + Wk (n)

(1) (q)

at the q-th receive antenna in the k-th subband. Here, Hk is the channel response of the q-th receive antenna in the k-th subband, Sk (n) is the n-th component of the SSU in the (q) k-th subband, Wk (n) is the n-th white noise component at the q-th receive antenna in the k-th subband, and Nk is the number of observations collected at an antenna in the k-th subband over (q) one sensing interval. We assume Hk depends on k and q, but is constant over a sensing interval. Note that Ni = Nj when Bi = Bj since the sampling rate is usually determined by the Nyquist rate. The spectrum sensing problem in the k-th subband can be formulated as a statistical hypothesis testing problem of choosing between “H0,k : The k-th subband is not currently

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being used” and “H1,k : The k-th subband is currently being used,” which can be expressed as  H0,k : Rk = W k , (2) H1,k : Rk = H k S Tk + W k . In (2), the superscript T denotes the transpose; the A × Nk matrix Rk = [Rk (1), Rk (2), . . . , Rk (Nk )] is the collection of all the observations in the k-th subband over the A receive (1) (2) antennas with the A×1 vector Rk (n) = [Rk (n), Rk (n),. . . , (A) Rk (n)]T = H k Sk (n) + W k (n) denoting the collection of the n-th observations in the k-th subband over the A receive antennas; the Nk × 1 vector S k = [Sk (1), Sk (2), . . . , Sk (Nk )]T is the collection of the SSU components of Rk ; the A × 1 (1)

(A) T

(2)

vector H k = [Hk , Hk , . . . , Hk ] is the collection of the channel responses in the k-th subband over the A antennas; and the A × Nk matrix W k = [W k (1), W k (2), . . . , W k (Nk )] is the collection of all the noise components in the k-th subband over the A receive antennas with the A × 1 vector W k (n) = (1)

(2)

(A)

T

[Wk (n), Wk (n), . . . , Wk (n)] denoting the collection of the n-th white noise components in the k-th subband over the A receive antennas. In practical cases, we have commonly Nk  A for k = 1, 2, . . . , K. We assume that the signal components {S k }K k=1 are widesense stationary and zero-mean and that the vectors S i and S j of signal components are independent of each other. As in most other studies, the elements in the noise matrix W k are assumed to be zero-mean, independent and identically distributed (i.i.d.) Gaussian random variables with variance 2 = Bk PW , where PW denotes the two-sided PSD of the σW k noise. That is, the noise is white within each subband. It is also assumed that the matrices W i and W j of noise components are independent of each other and that {W k }K k=1 are independent . of {S k }K k=1 III. P ROPOSED D ETECTION S CHEME In the hypothesis testing problem of H0,k versus H1,k , we assume that the CR is provided with neither the variances of the SSU and noise nor the channel responses {H i }K i=1 . Then, we can regard the detection problem as that of a hypothesis testing in the presence of unknown parameters. Denote by M i (n1 , n2 ) = E{Ri (n1 )RH i (n2 )} the A × A covariance matrix of Ri (n1 ) and Ri (n2 ), where E{·} and the superscript H denote the expectation operator and conjugate transpose, respectively. Then, we have M i (n1 , n2 ) = 2 I δ under H0,i , and σW i A n1 −n2   2 M i (n1 , n2 ) = E Si (n1 )SiH (n2 ) H i H H i + σWi I A δn1 −n2 (3) under H1,i , where δm = 1 if m = 0 and 0 if m = 0 is the Kronecker delta function and I m is the m × m identity matrix. The model for the spatial correlation between observations in this paper is implicitly defined by (3). When the characteristics of the SSU are unknown to the CR, it is commonly assumed [10]–[12] that the zero-mean signal component S i is a vector of i.i.d. complex Gaussian variables

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with variance σi2 , which is readily justifiable particularly when the spectrum user employs orthogonal frequency division multiplexing, the channel is a flat-fading channel, and the sampling rate is equal to the Nyquist rate [13]. Even when the Gaussian assumption is not practically appropriate, the Gaussian model in many cases leads to tractable analysis and useful detectors [12], [13], [16]. Under the Gaussian assumption, we have E{Si (n1 )SiH (n2 )} = σi2 δn1 −n2 in (3). A. Joint Probability Density Functions of {R1 , R2 , . . . , RK } Let r i = [ri (1), r i (2), . . . , ri (Ni )] be a realization of Ri , (1)

(2)

fRi (r i |H0,k ) with respect to PW and M i (n, n), respectively, we can obtain the estimate M i (n, n) = Bi Δ(r i ) of M i (n, n) for i ∈ B \ (BV ∪ {k}) and the estimate  1   P Ni tr {Δ(r i )} W,0 = ANi i∈BV ∪{k}

of PW under H0,k , a generalized version of the conventional maximum likelihood (ML) result in [10]. Similarly, the joint pdf of {R1 , R2 , . . . , RK } under H1,k can be expressed as fR1 ,R2 ,...,RK (r 1 , r 2 , . . . , r K |H1,k )   − ANi  N tr{Δ(r )} i i exp − π i∈B PW =

fR1 ,R2 ,...,RK (r 1 , r 2 , . . . , r K |H0,k )  N  i H −1  exp −n=1 ri (n)M i (n, n)r i (n) = π ANi [det {M i (n, n)}]Ni i∈BV ∪{k}

·

i∈B\(BV ∪{k}) −

π =







ANi

exp −

i∈B

i∈BV ∪{k}

·

·

i∈BV ∪{k}

BiANi

Ni

Ni tr{Δ(r i )} PW









ANi

i∈BV \{k}

BiANi

PW



  exp −Bi Ni tr M −1 i (n, n)Δ(r i )

i∈B\(BV \{k})

[det {M i (n, n)}]Ni

. (7)

As  shown in Appendix A, by maximizing the joint pdfs j∈BV \{k} fRj (r j |H1,k ) and fRi (r i |H1,k ) with respect to PW and M i (n, n), respectively, we can obtain the estimate







i (n, n)}]

i∈BV \{k}



i∈BV \{k}

 N  i H exp − ri (n)M −1 (n, n)r (n) i i n=1 π ANi [det {M

(6)

i∈BV ∪{k}

T

(A)

where r (n) = [ri (n), ri (n), . . . , ri (n)] . Then, since Ni Hi −1 −1 n=1 r i (n)M i (n, n)r i (n) = Bi Ni tr{M i (n, n)Δ(r i )}, the joint probability density function (pdf) of {R1 , R2 , . . . , RK } under H0,k can be expressed as



(5)

M i (n, n) = Bi Δ(r i )

(8)

ANi

i∈BV ∪{k}

of M i (n, n) for i ∈ B \ (BV \ {k}) and the estimate

PW



  exp −Bi Ni tr M −1 i (n, n)Δ(r i )

i∈B\(BV ∪{k})

[det {M i (n, n)}]Ni

 P W,1 = , (4)

where Δ(r i ) = 1/(Bi Ni )r i r H i , det (·) and tr(·) denote the determinant and trace of a matrix, respectively, and the subset BV = {b1 , b2 , . . . , bKV } of B = {1, 2, . . . , K} denotes the set of the indices of vacant subbands with KV denoting the number of vacant subbands or the cardinality |BV | of BV . Among the three unknown quantities PW , BV , and {M i (n, n) : i ∈ B \ (BV ∪ {k})} in (4), let us first estimate PW and {M i (n, n) : i ∈ B \ (BV ∪ {k})}. Since {M i (n, n) : i ∈ B \ (BV ∪ {k})} and PW are generally dependent on each other, it seems not possible to obtain the optimal estimates of PW and {M i (n, n) : i ∈ B \ (BV ∪ {k})}. We will instead obtain suboptimal estimates of PW and M i (n, n) from {Rj }j∈BV ∪{k} under H0,k and from Ri , respectively, for i ∈ B \ (BV ∪ {k}) by noting that there exist only noise components in the vacant subbands BV ∪ {k} under H0,k and the covariance matrix M i (n, n) is determined by the distributions of the observations Ri . Specifically,  as shown in Appendix A, by maximizing the joint pdfs j∈BV ∪{k} fRj (r j |H0,k ) and





1

i∈BV \{k}

ANi

Ni tr {Δ(r i )}

(9)

i∈BV \{k}

of PW under H1,k . Let us mention that (5) and (8), though not ML estimates, are of the same expressions as the ML estimates in other system models [10]. As described in Appendix A, since  P W,t and M i (n, n) are on the average the same as PW and  M i (n, n), respectively, the estimates P W,t and M i (n, n) are consistent [22] estimates. Next, the set BV in (6) and (9) is still unknown and needs to be estimated. Unfortunately, the optimum estimation requires considerably complicated procedure, if not impossible. Thus, based on the observations that KV ≥ 1 with high probability and that the subband with a lower PSD is more likely to contain only noise than that with a higher PSD, we estimate BV in an intuitive way. In particular, since a significantly adverse consequence could occur if an occupied subband is erroneously regarded as a vacant subband, we take a rather conservative path   by adopting the estimate K V = |BV | = 1 of |BV |, with which only the subband with the lowest PSD shall be estimated to be vacant. In other words, without requiring the information on the number of the vacant subbands, in addition to avoiding a significantly adverse consequence, only the subband which is most

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

likely to be vacant is regarded as the vacant subband. Specifically, by noting that the sum of PSDs of  observations over i 2 A antennas in the i-th subband is 1/(Bi Ni ) N n=1 r i (n) = tr{Δ(r i )}, we employ the estimate B (10) V = {θ} of BV , where θ = arg mini∈B tr{Δ(r i )}. Now, based on the pdf (35) shown in Appendix A, the likelihood ratio can be expressed as (11), shown at the bottom of the page.

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information on the SSU, it is very difficult to obtain the detection scheme by employing the likelihood functions as in the case of temporally independent observations. We will thus obtain a detection scheme based on the covariance matrix of a number of consecutive observations as in [8] and [9] when the observations are temporally dependent. DenotT ing by Ri (n; L) = [RTi (n), RTi (n + 1), . . . , RTi (n + L − 1)] the vector of AL consecutive observations with L called the window size, the AL × AL covariance matrix M i (L) = E{Ri (n; L)RH i (n; L)} of Ri (n; L) can be estimated via  M i (L) = Bi Δ(r i ; L),

B. Test Statistics of the Proposed Scheme After some manipulations, the test statistic in the k-th subband can be obtained from (11) as

as in [8], where

Tk (r 1 , r 2 , . . . , r K ) =

[Nθ tr {Δ(r θ )} + Nk tr {Δ(r k )}]

Δ(Ri ; L) =

Nθ +Nk

[tr {Δ(r θ )}]Nθ [det {Δ(r k )}]

Nk A

.

(12)

Note that the test statistic (12) is valid for both the cases of k = θ and k = θ and that the test statistic (12) for k = θ is equivalent to the test statistic of the AGM scheme. We would  also like to mention that the estimate K V of the number KV of vacant subbands is used only when the test statistic (12) is obtained, but not when (12) is used in determining whether the k-th subband is occupied or not. Once the observations in K subbands are available, the test statistic (12) can be computed and employed for WSS regardless of the actual number of vacant subbands. Recollecting that the estimate θ denotes the subband with the lowest PSD among the K subbands, the quantity Bθ tr{Δ(r θ )} = (1/Nθ )tr{r θ r H θ } is expected to indicate the sum of noise powers over A antennas in a (vacant) subband. In other words, tr{Δ(r θ )} in the test statistic (12) is a measure of the PSD of the noise except for the unlikely case of KV = 0. On the other hand, the quantity tr{Δ(r k )} is the sum of PSDs of observations over A antennas in the k-th subband. In the meantime, det{Δ(r k )} possesses some information on the correlation among observations across A antennas in the k-th subband. In short, the test statistic Tk (r 1 , r 2 , . . . , r K ) makes use of the PSD of noise via tr{Δ(r θ )}, PSDs of observations in the k-th subband via tr{Δ(r k )}, and correlations among observations in the k-th subband via det{Δ(r k )}. Remark: In addition to spatial correlation, the signal components in a subband at the receiver end could frequently possess correlation in time because of oversampling or the dispersiveness of channels [8], [9]. When the observations are temporally correlated, if we do not have the

(13)

1 B i Ni

Ni 

Ri (n; L)RH i (n; L)

(14)

n=−L+2

is the sample covariance matrix of Ri (n; L) divided by Bi with (q) Ri (n) = 0 for n ∈ {1, 2, . . . , Ni }. Replacing the estimates of M θ (n, n) and M k (n, n) in the test statistic (12) with those of M θ (L) and M k (L), respectively, we will get the test statistic Gk (r 1 , r 2 , . . . , r K ; L) =

[Nθ tr {Δ(r θ ; L)} + Nk tr {Δ(r k ; L)}]Nθ +Nk Nk

[tr {Δ(r θ ; L)}]Nθ [det {Δ(r k ; L)}] AL

(15)

in the k-th subband when the observations are temporally dependent. Although the proposed scheme (15) employs the sample covariance matrix (14) of a number of consecutive observations as in the schemes in [8] and [9], it is clearly distinct from the schemes in [8] and [9] in that, to obtain the test statistic in the k-th subband, all the observations (r 1 , r 2 , . . . , r K ) over the K subbands are used while only the observation r k is used in [8] and [9]. Note that no specific model of the temporal correlation is assumed in deriving the test statistic (15).  It is obvious from (12) and (15) that the test statistic (15) includes the test statistic (12) as a special case of L = 1 since Δ(r i ; 1) = Δ(r i ). Thus, from now on, we will concentrate on the test statistic (15) assuming that Nk = NO for k = 1, 2, . . . , K for a simplicity reason. The test statistic in the k-th subband can then be rewritten as Gk (r 1 , r 2 , . . . , r K ; L) =

[tr {Δ(r θ ; L)} + tr {Δ(r k ; L)}]2 1

tr {Δ(r θ ; L)} [det {Δ(r k ; L)}] AL

.

(16)

fR1 ,R2 ,...,RK (r 1 , r 2 , . . . , r K |H1,k )|B

   V =B V ,PW =PW,1 ,M i (n,n)=M i (n,n) fR1 ,R2 ,...,RK (r 1 , r 2 , . . . , r K |H0,k )|B =B,P =P ,M (n,n)=M V V W W,0 i i (n,n) ⎧ [tr{Δ(r k )}]ANk ⎪ k = θ, ⎨ AANk [det{Δ(r )}]Nk , k =  1 [N tr{Δ(r )}+N tr{Δ(r )}]A(Nθ +Nk ) θ k k ⎪ ⎩ Nθ +NkAN θ , k = θ AN N A

k [tr{Δ(r θ )}]

θ [det{Δ(r k )}]

k

(11)

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TABLE I C OMPUTATIONAL C OMPLEXITY OF S EVERAL D ETECTION S CHEMES IN WSS

4AL. Consequently, we will have ηk ≥ 4AL. As shown in Appendix B, the probability Pr{Gk (R1 , R2 , . . . , RK ; L) ≤ ηk |Ht,k } can be rewritten as     Pr Gk (R1 , R2 , . . . , RK ; L) ≤ ηk Ht,k ηk y

∞ 4

{1 − FXθ (ak (x, y)|Xk = x, Ht,k )}

= 0

· fXk ,Yk (x, y|Ht,k ) dx dy

C. Computational Complexity As the number of multiplications is a useful index indicating the computational complexity of a processor [9], [11], let us investigate the complexity of the proposed scheme in comparison with those of other schemes in terms of the number of multiplications. Assume that one complex multiplication is equivalent to four real multiplications (RMs) and that the m-th root has the same computational complexity as m − 1 RMs. In the proposed scheme, since {Δ(r i ; L)}K i=1 are Hermitian block Toeplitz when A, L = 2, 3, . . . [9], we need K[2ANO (AL + L − 1) + (AL/2){A(3 − 2L) + 1 − 2L} + L + 1] RMs to compute {Δ(r i ; L)}K i=1 from {r 1 , r 2 , . . . , r K } in the K subbands. Next, computing the determinants of {Δ(r i ; L)}K i=1 requires O(A3 KL3 ) RMs, and computing {Gi (r 1 , r 2 , . . . , r K ; L)}K i=1 requires K(AL + 2) RMs. Thus, in terms of RMs, the computational complexity of the proposed scheme is O(2A2 KLNO + A3 KL3 ). Table I compares the computational complexity of several schemes in WSS. When the AGM, CAV, MES, or MME scheme is adopted in WSS, it is assumed in this section and Section V that the K subbands are searched by employing a bank of K identical detectors, each of which is dedicated to search one subband. It is observed that the ED scheme has a lower computational complexity than other detection schemes. Since NO  A in most cases, the proposed, AGM, CAV, MES, and MME schemes have almost the same computational complexity when L = 1. In addition, when L = 2, 3, . . ., the computational complexity of the proposed scheme is the same as that of the MME scheme, while it is slightly higher than that of the CAV scheme and L times that of the AGM and MES schemes. IV. P ERFORMANCE A NALYSIS When the threshold of the proposed sensing scheme (16) is ηk , the false alarm and detection probabilities of the proposed scheme in the k-th subband can be expressed as     PF A,k (ηk ; L) = 1−Pr Gk (R1 , R2 , . . . , RK ; L) ≤ ηk H0,k (17) and

    PD,k (ηk ; L) = 1−Pr Gk (R1 , R2 , . . . , RK ; L) ≤ ηk H1,k , (18)

respectively. Here, it can be shown that the test statistic Gk (R1 , R2 , . . . , RK ; L) is always larger than or equal to

0

(19)

for t = 0 and 1 when AL = 2, 3, . . ., where fXk ,Yk (x, y|Ht,k ) is the joint pdf of Xk =

1 tr {Δ(Rk ; L)} PW

(20)

and Yk =

1 1 [det {Δ(Rk ; L)}] AL PW

(21)

under Ht,k , FXθ (z|Xk = x, Ht,k ) is the conditional cumulative distribution function (cdf)  of Xθ given Xk = x under Ht,k , and ak (x, y) = (ηk y − 2x − ηk2 y 2 − 4ηk xy)/2. Note that  O 2 2 (20) can be written as Xk = L N n=1 Rk (n) /(NO σWk ) and that the quantity Xθ is the first order statistic [23] X[1] of {Xi }K i=1 . A. Performance When the Observations are Spatially Correlated and Temporally Independent in Nonfading Channels When L = 1 and the zero-mean signal component S i is a vector of i.i.d. complex Gaussian random variables with variance σi2 for i = 1, 2, . . . , K, the distribution of 2NO Xi is central chi-square with degree of freedom 2ANO under H0,i and the distribution of Xi can be approximated as a Gaussian distribution with mean A(γi + 1) and variance (A/NO )(Aγi2 + 2 ) 2γi + 1) under H1,i [12], [24], where γi = H i 2 σi2 /(AσW i is the signal to noise ratio (SNR) in the i-th subband. Based on the distributions of {Xi }K i=1,i=k , we can obtain the probability 1 − FXθ (ak (x, y)|Xk = x, Ht,k ) as described in Appendix C. Next, when A = 2, 3, . . ., the matrix (NO /PW )Δ(Rk ; 1) = 2 )Rk RH (1/σW k in (20) and (21) is an uncorrelated central k Wishart matrix under H0,k and a correlated central Wishart H 2 2 2 )E{Rk (n)RH matrix with (1/σW k (n)} = H k H k (σk /σWk ) + k I A under H1,k [25], based on which we can obtain the joint pdf fXk ,Yk (x, y|Ht,k ) as described in Appendix D. With these two results, we can compute PF A,k (ηk ; 1) and PD,k (ηk ; 1) using (19). For example, when A = 2, we have the joint pdfs fXk ,Yk (x, y|H0,k ) =

2 

2NO2NO

y 2NO −3



x2 − 4y 2 e−NO x

(NO − i)!

i=1

(22)

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

for 0 < 2y ≤ x and fXk ,Yk (x, y|H1,k ) =

For example, when A = 2 and L = 1 in fading channels, we have the false alarm probability

2NO2NO −1 y 2NO −3 2  γk (1 + 2γk )NO −1 (NO − i)!

ηk y

∞ 4 PF A,k (ηk ; 1) ≈ 1 −

i=1

    NO γ k  2 NO (1 + γk )x · exp − x − 4y 2 sinh 1 + 2γk 1 + 2γk

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∞ 

I0,k (x, y) dx dy

(24)

I1,k (x, y) dx dy

(25)

and the detection probability ηk y

∞ 4 0 2y

from (17) and (18), respectively, using (19), (22), (23), and (41) with A = 2, where I0,k (x, y) is the product of the righthand sides of (22) and (41), and I1,k (x, y) is the product of the right-hand  sides of (23) and (41). Here, when K increases, the term K i=1 {(Γ(2NO , NO ak (x, y))/Γ(2NO )) Pr(H0,i ) + i=k  √ Q( NO {ak (x, y) − 2(γi +1)}/ 4γi2 +4γi +2) Pr(H1,i )} in (41) decreases since Γ(x, α)/Γ(x) ≤ 1 for nonnegative α, Q(x) ≤ 1, and Pr(H0,i ) + Pr(H1,i ) = 1, which implies that the false alarm and detection probabilities in (24) and (25) will be higher when K is larger. When the SNRs {γi }K i=1,i√ =k are low, using (Γ{2NO , N O ak (x, y)})/(Γ(2NO )) ≈ Q[( NO {ak (x, y) − 2(γi + 1)})/ ( 4γi2 + 4γi + 2)] as discussed in Appendix C, the false alarm probability (24) can be further simplified into ηk y

PF A,k (ηk ; 1) ≈ 1 −

Γ {2NO , NO ak (x, y)} Γ(2NO )

!K−1

PD,k (ηk ; 1) ≈ 1 −

· fXk ,Yk (x, y|H0,k ) dx dy

(26)

and the detection probability (25) can be simplified into ηk y

PD,k (ηk ; 1) ≈ 1 − 0

Γ {2NO , NO ak (x, y)} Γ(2NO )

!K−1

2y

· fXk ,Yk (x, y|H1,k ) dx dy.

"∞"∞ "∞  where It,k (x, y) = 0 0 · · · 0 It,k (x, y){ K i=1 fγi (ui )} du1 du2 . . . duK for t = 0 and 1 with fγi (u) denoting the pdf of the SNR γi in the i-th subband. In the case of Rayleigh fading, the pdf fγi (u) = (1/Γ(A))(A/γi )A uA−1 exp(−(A/γi )u) can be inserted in (28) and (29) for the evaluation of the false 2 alarm and detection probabilities, where γi = E(γi ) = σi2 /σW i is the average SNR in the i-th subband [5]. When the SNRs {γi }K i=1,i=k are low, (28) can be simplified as ηk y

∞ 4

Γ {2NO , NO ak (x, y)} Γ(2NO )

PF A,k (ηk ; 1) ≈ 1 −

!K−1

0 2y

· fXk ,Yk (x, y|H0,k ) dx dy

(30)

and (29) can be simplified as ηk y

∞ ∞ 4 PD,k (ηk ; 1) ≈ 1 − 0

0

Γ {2NO , NO ak (x, y)} Γ(2NO )

!K−1

2y

· fXk ,Yk (x, y|H1,k )fγk (u) dx dy du.

(31)

In the asymptotic case of NO → ∞, we have derived in Appendix E a simpler expression of the false alarm probability PF A,k (ηk ; 1) when the SNRs {γi }K i=1,i=k are low. V. S IMULATION R ESULTS

0 2y

∞ 4

(29)

ηk y

∞ 4 0 2y

0 2y

∞ 4

I1,k (x, y) dx dy,

and the detection probability

ηk y 4

PD,k (ηk ; 1) ≈ 1 −

(28)

(23)

for 0 < 2y ≤ x of Xk and Yk . Then, we have the false alarm probability

PF A,k (ηk ; 1) ≈ 1 −

I0,k (x, y) dx dy 0 2y

(27)

Let us now evaluate and compare the detection performances of several schemes when the observations are spatially correlated and temporally independent/dependent through simulations in fading channels. In the simulations, it is assumed that the i-th subband is occupied with probability Pr(H0,i ) = 0.5 for i = 1, 2, . . . , K and the vector H i of channel responses in the i-th subband is composed of i.i.d. complex Gaussian random variables with mean zero and variance one. We also assume Ni = NO and γi = γ for i = 1, 2, . . . , K.

B. Performance When the Observations are Spatially Correlated and Temporally Independent in Fading Channels

A. Case 1: Spatially Correlated But Temporally Independent Observations

By averaging PF A,k (ηk ; L) and PD,k (ηk ; L) with respect to {γi }K i=1 , we can obtain the false alarm and detection probabilities, respectively, of the proposed scheme in fading channels.

Assuming that the zero-mean signal components S i are i.i.d. complex Gaussian random variables with variance σi2 , we first address the detection performances when the observations are

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Fig. 2. The detection probability of the proposed scheme when γ = −6 dB, the false alarm probability is 0.1, and the observations are temporally independent. TABLE II T HE P ROBABILITY T HAT THE S UBBAND W ITH THE L OWEST PSD I S UNDER H0,k W HEN Pr(H0,k ) = 0.5 AND γk = −6 dB FOR k = 1, 2, . . . , K

Fig. 1. The false alarm and detection probabilities of the proposed scheme when A = 2, K = 4, and the observations are temporally independent. (a) γ = −16 dB. (b) γ = −2 dB.

spatially correlated but time-wise independent. In the proposed, MME, and CAV schemes, we set L = 1. Fig. 1 shows the false alarm and detection probabilities of the proposed scheme as a function of the threshold ηk for γ = −16 dB and −2 dB when A = 2 and K = 4. It is clearly observed that the analytical results, calculated numerically from (28) and (29), agree closely with the simulation results. We can also observe that the simpler expressions (30) and (31) are closer to (28) and (29) when the SNR is low than when the SNR is high. Let us next address the influence of the number K of subbands on the detection probability of the proposed scheme. In Fig. 2, we have shown the detection probability of the proposed scheme as a function of the number K of subbands for four combinations of the numbers (A and NO ) of antennas and observations. It is observed that the detection probability of the proposed scheme gets higher, although rather slow, as the number K of subbands increases. A possible explanation for this observation is as follows. In Appendix F, the probability that the subband with the lowest PSD is under the null hypothesis is derived. As shown in Table II, the probability gets higher when the number K of subbands increases, eventually

producing better estimate of the noise PSD. This allows the proposed scheme to exhibit better detection performance when K is larger. Let us just mention that the asymptotic behaviour of the probability is discussed in [26]. Fig. 3 shows the receiver operating characteristics (ROCs) of several schemes for A = 1, 2, and 4, respectively. In obtaining Fig. 3(a), the DIM scheme is assumed to have the estimate M of the minimum number of the vacant subbands in the wideband channel. It should be noted that the DIM scheme cannot be employed at the CR with A = 2, 3, . . . and the AGM, CAV, MES, and MME schemes cannot be employed at the CR with A = 1. As a means to consider noise uncertainty in the proposed and ED schemes, we have assumed that the noise 2 2 variance is estimated as σ# Wi = μi σWi for the proposed and ED schemes, where the noise uncertainty factor 10 log10 μi is assumed to be distributed uniformly in the interval [−β, β]. In practice, it is reported that β is normally 1–2 dB [6], [11]. The noise uncertainty is not taken into account in the AGM, CAV, DIM, MES, and MME schemes since these schemes are all known to be robust to noise uncertainty [8]–[11], [16]. In these figures, the proposed scheme is clearly observed to be robust to noise uncertainty. When A = 1, the proposed scheme provides better detection performance than the ED and DIM with M = 10, but the DIM with M = 2 and 6 provide slightly better detection performance than the proposed scheme. On the other hand, when A = 2 and 4, it is observed that the proposed scheme exhibits better detection performances than other schemes. We can also observe that the proposed scheme provides more significant performance gains over the AGM,

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

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Fig. 4. The detection probability of the proposed scheme when K = 16, γ = −6 dB, the false alarm probability is 0.1, and the observations are temporally dependent.

Specifically, generated as

the

signal

components

O {Si (n)}N n=1

Si (n) = ρSi (n − 1) + Zi (n)

Fig. 3. The ROCs of various detection schemes for WSS when K = 16, NO = 50, γ = −6 dB, and the observations are temporally independent. (a) A = 1. (b) A = 2. (c) A = 4.

CAV, MES, and MME schemes when the number of antennas is smaller and that the performance gain of the proposed scheme over the ED scheme becomes larger as the number of antennas increases. B. Case 2: Spatially Correlated and Temporally Dependent Observations To describe temporally dependent observations, we have employed the first-order autoregressive model as in [27].

are

(32)

for i = 1, 2, . . . , K, where the signal dependence parameter ρ determines the degree of temporal correlation among the signal components with |ρ| < 1, the initial value Si (1) is set to a complex Gaussian random variable with mean zero and O variance σi2 , and {Zi (n)}N n=2 are i.i.d. complex Gaussian ran2 = (1 − ρ2 )σi2 . dom variables with mean zero and variance σZ i O Thus, assuming that Si (1) is independent of {Zi (n)}N n=2 , we have E{Si (n1 )SiH (n2 )} = σi2 ρ|n1 −n2 | . Let us first look into the influence of the window size L and signal dependence parameter ρ on the detection performance of the proposed scheme with Fig. 4, in which the detection probability of the proposed scheme is shown as a function of the window size L for several combinations (|ρ|, A, NO ) of the signal dependence parameter, number of antennas, and number of observations. It is observed that the detection probability of the proposed scheme (i) decreases as L gets larger when |ρ| = 0.1, (ii) is higher with L = 1 or 2 than with L = 3, 4, . . . when |ρ| = 0.5, and (iii) is higher with L = 2, 3, 4, 5 than with L = 1 when |ρ| = 0.9. An implication of these observations is that, unless the value of |ρ| is close to 1, it suffices to set the window size L to 1 in the detection of correlated signals also. If |ρ| is close to 1, we might consider increasing the window size to 2. Fig. 5 shows the ROCs of several schemes for various values of |ρ| and L. Since the proposed scheme with L = 1 has been observed to be robust to noise uncertainty in Fig. 3, noise uncertainty is now taken into account for the ED scheme only. It is again observed clearly that the proposed scheme provides better detection performance than other detection schemes. In addition, we can observe that the proposed scheme generally provides the best performance with L = 1 except when |ρ| is close to 1 and further improved detection performance with a larger window size when |ρ| is close to 1, which we have observed in Fig. 4 also. Note that the proposed scheme with

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schemes as |ρ| increases, incurring possibly higher computational/hardware complexity. VI. C ONCLUDING R EMARK We have proposed novel detection schemes of WSS for cognitive radio with multiple receive antennas, providing reasonable performance improvement when the observations are spatially correlated and temporally independent/dependent. Taking the observations in all subbands into consideration for sensing a subband, the test statistics of the proposed schemes are functions of the sample covariance matrix in the subband under consideration and that in the subband exhibiting the lowest PSD. The detection performance of the proposed schemes is observed to improve more as the number of subbands increases. The detection performances of several schemes have been evaluated and compared when the observations exhibit spatial correlation and temporal independence and when they exhibit spatial correlation and temporal dependence. When the observations are temporally independent, the proposed scheme is shown not only to outperform other schemes but also to be robust to noise uncertainty. It has been observed clearly that the proposed scheme provides better detection performance than other schemes for temporally dependent observations also. It is noteworthy that the proposed scheme with window size one can be used rather universally over a large span of the degree of temporal correlation among signal components and that, when the degree of temporal correlation is close to one, some additional gain can be secured with window size two. A PPENDIX A E STIMATES AND L IKELIHOOD F UNCTIONS Estimates of PW  and M i (n, n): Taking the natural logarithms of the pdf i∈Vt fRi (r i |Ht,k ), we have the log likelihood function % $   fRi (r i |Ht,k ) = − ANi ln(π) ln i∈Vt

−

i∈Vt

 i∈Vt

Fig. 5. The ROCs of various detection schemes in temporally dependent observations when A = 2, K = 16, NO = 50, and γ = −6 dB. (a) |ρ| = 0.1. (b) |ρ| = 0.5. (c) |ρ| = 0.9.

L = 1 performs better than other detection schemes with L = 1 even when |ρ| is close to 1. An interesting consequence of these observations is that the proposed scheme with L = 1 is quite a useful tool also in detecting temporally dependent signals when the degree of dependence (or correlation) among signal components is unknown to the CR. On the other hand, the detection performances of the CAV and MME schemes are in general more dependent on the window size L, and a larger value of L is more preferable for the CAV and MME

ANi ln(Bi PW ) −

1  Ni tr {Δ(r i )} PW

(33)

i∈Vt

for t = 0 and 1, where ln(·) denotes the natural logarithm, V0 = BV ∪ {k}, and V1 = BV \ {k}. Then, taking the derivative of (33) with respect to PW , we have $ %  ∂ ln fRi (r i |Ht,k ) ∂PW i∈Vt  ANi 1  i∈V =− t + 2 Ni tr {Δ(r i )} , (34) PW PW i∈Vt

from which we can obtain the estimates (6) and (9) of PW by finding the value of PW which makes (34) equal to zero. Next, based on the property that M i (n, n) is positive definite Hermitian and using that [28] (∂tr{M −1 i (n, n)Δ(r i )})/

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

−1 −1 T (∂M −1 i (n, n)) = Δ (r i ) and (∂ det{M i (n, n)})/(∂M i −1 T (n, n)) = det{M i (n, n)}M i (n, n), the estimates (5) and (8) of M i (n, n) can be obtained [10]. In short, the estimate M i (n, n) is equal to the sample covariance matrix of r i (n). Let us note that, in our system model, it is not mathematically feasible to obtain the estimates of H i and σi2 = E{Si (n)SiH (n)} separately for the estimation of M i (n, n).  ANi ) We would like to add that, since P W,t = (1/ t  N i   i∈V Ni 2 i∈Vt n=1 (|r i (n)| /Bi ) = (1/ i∈Vt ANi ) i∈Vt n=1 √ 2 A (q)  |wi (n)/ Bi | under Ht,k and M i (n, n) = (1/Ni ) q=1  Ni Ni H n=1 r i (n)r i (n) = (1/Ni ) n=1 {H i si (n) + w i (n)}{H i si (n) + wi (n)}H when the i-th subband is occupied, the quan tities P W,t and M i (n, n) on the average will be the same as the noise PSD PW and covariance matrix M i (n, n) in (3), (1) (2) respectively, where si (n) and wi (n) = [wi (n), wi (n),. . . , (A) wi (n)]T denote realizations of Si (n) and W i (n), respectively. In other words, the estimates (5), (6), (8), and (9) are consistent estimates [22]. Likelihood functions with the estimates: Substituting B V,  P , and M (n, n) obtained in (5), (6), and (8)–(10) into (4) W,t i and (7), we have the joint pdf

Then, we can obtain the probability Pr{Gk (R1 , R2 , . . . , RK ; L) ≤ ηk |Ht,k } for t = 0 and 1 as     Pr Gk (R1 , R2 , . . . , RK ; L) ≤ ηk  Ht,k

=

Pr 0

   (Xθ + Xk )2  ≤ ηk  Xk = x, Yk = y, Ht,k Xθ Y k

0

× fXk ,Yk (x, y|Ht,k ) dx dy ∞∞ =

  Pr Xθ2 − (ηk y − 2x)Xθ + x2 ≤ 0

0 0

Xk = x, Ht,k } fXk ,Yk (x, y|Ht,k ) dx dy ηk y

∞ 4

{ FXθ (bk (x, y)| Xk = x, Ht,k )

= 0

0

−FXθ ( ak (x, y)| Xk = x, Ht,k )} · fXk ,Yk (x, y|Ht,k ) dx dy ηk y

  × (r 1 , r 2 , . . . , r K |Ht,k ) ⎛ = (πe)−NB ⎝ ⎛



∞ ∞

∞ 4

fR1 ,R2 ,...,RK

˜

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i∈V(t

{1 − FXθ (ak (x, y)|Xk = x, Ht,k )}

=    B V =B V ,PW =P W,t ,M i (n,n)=M i (n,n)

⎞⎡

Bi−ANi ⎠ ⎣

⎤−N˜t

 Ni tr {Δ(r i )} ⎦ ˜t N i∈V(t ⎞

⎟ ⎜  ·⎝ [det {Bi Δ(r i )}]−Ni ⎠ i∈B\V(t

(35)

˜B= ANi , N ˜t = under Ht,k for t = 0 and 1, where N i∈B    ANi , V0 = {θ, k}, and V1 = {θ}\{k}. In obtaining (35), i∈V(t −1 −1 we have used Bi Ni tr{M i (n, n) Δ(r i )} = Ni tr{Δ (r i ) Δ(r i )} = ANi .  ˜1 = 0 and P  In (35), since N (Ni tr{Δ(r i )}/ W,1 = i∈V(1  ˜1 ) = 0/0, we have [ ˜1 )]−N˜1 = (0/0)0 N (Ni tr{Δ(r i )}/N i∈V(1 when t = 1 and k = θ. Now, the noise PSD is nonzero and finite 0  in practical situations, and thus we can set P W,1 = 1. In other words, we assume (0/0)0 = 1 for t = 1 and k = θ in (35).

0

· fXk ,Yk (x, y|Ht,k ) dx dy, (37)  where ak (x, y) = (1/2)(ηk y  − 2x − ηk2 y 2 − 4ηk xy) and bk (x, y) = (1/2)(ηk y − 2x + ηk2 y 2 − 4ηk xy). Note that 0 < ak (x, y) < x and x < bk (x, y) < ηk y in the interval 0 < x < (ηk y/4) of integration in (37). When Xk = x is given, since Xθ is the minimum of {X1 , X2 ,. . . ,Xk−1 , x, Xk+1 , Xk+2 ,. . . ,XK } whose elements are independent of Yk , it is straightforward to see that Xθ is independent of Yk . Therefore, the condition Yk = y has been deleted when the fourth and fifth lines are obtained from the second and third lines in (37). We also have FXθ (bk (x, y)|Xk = x, Ht,k ) = 1 since Xθ < bk (x, y) is true from Xθ ≤ x and bk (x, y) > x when Xk = x and 0 < x < (ηk y/4), which have been taken into account in obtaining the last two lines from the fifth last, fourth last, and third last lines in (37). A PPENDIX C A PPROXIMATING THE P ROBABILITY 1 − FXθ (ak (x, y)|Xk = x, Ht,k ) Under H0,i , since 2NO Xi is chi-square with degree of freedom 2ANO [12], the cdf of Xi is

A PPENDIX B D ISTRIBUTION OF Gk (R1 , R2 , . . . , RK ; L) The test statistic Gk (R1 , R2 , . . . , RK ; L) in (16) can be rewritten as (Xθ + Xk )2 Gk (R1 , R2 , . . . , RK ; L) = . Xθ Y k

0

(36)

FXi (x|H0,i ) = 1 −

Γ(ANO , NO x) , Γ(ANO )

(38)

"∞ where Γ(u, v) = v xu−1 e−x dx for u > 0 is the upper incomplete gamma function and Γ(u) = Γ(u, 0) is the gamma function. Under H1,i , since Xi is approximately Gaussian with

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mean A(γi + 1) and variance (A/NO )(Aγi2 + 2γi + 1) [24], we have the cdf

√ NO {x − A(γi + 1)} (39) FXi (x|H1,i ) ≈ 1 − Q  A (Aγi2 + 2γi + 1) √ "∞ of Xi , where Q(u) = (1/ 2π) u exp(−x2 /2)dx. Now, recollecting that ak (x, y) < x when 0 < x < (ηk y/4) as discussed in Appendix B and that Xθ = X[1] , the probability 1 − FXθ (ak (x, y)|Xk = x, Ht,k ) in (19) can be rewritten as 1 − FXθ (a   k (x, y)|Xk = x, Ht,k ) = Pr X[1] > ak (x, y) Xk = x, Ht,k = Pr {All of {Xi }i∈B are larger than ak (x, y)| Xk = x, Ht,k } K  [{1 − FXi (ak (x, y) |H0,i )} Pr(H0,i ) = i=1 i=k

+ {1 − FXi ( ak (x, y)| H1,i )} Pr(H1,i )] .

(40)

Using (38) and (39) in (40), we have 1−FXθ (ak (x, y) |Xk = x, Ht,k )

K  Γ {ANO , NO ak (x, y)} Pr(H0,i ) ≈ Γ(ANO ) i=1

 (Xk − Xk2 − 4Yk2 ) from Xk = (1/NO )(Λ1 + Λ2 ) and Yk = √ (1/NO ) Λ1 Λ2 , and the Jacobian of the transformation from  (Xk , Yk ) to (Λ1 , Λ2 ) is J(Λ1 , Λ2 ) = (2NO2 Yk / Xk2 − 4Yk2 ). The joint pdf fXk ,Yk (x, y|Ht,k ) of Xk and Yk when AL = 2 can thus be expressed as fXk ,Yk (x, y|Ht,k ) =   · fΛ1 ,Λ2

2NO2 y x2 − 4y 2

2  NO 1 x + x2 − 4y 2 , 2  2  NO 1 x − x2 − 4y 2  Ht,k 2

(44)

for 0 < 2y ≤ x, where fΛ1 ,Λ2 ,...,ΛAL (·|Ht,k ) is the joint pdf of (Λ1 , Λ2 , . . . , ΛAL ) under Ht,k . When AL = 3, 4, . . ., letting (Z3 , Z4 , . . . , ZAL ) = (Λ3 , Λ4 , . . . , ΛAL ), we can similarly obtain Λ1 = 1/(2C1 )[C1 (NO Xk − C2)+{C12 (C2−NO Xk )2 −4C1 (NO Yk )AL}1/2 ]andΛ2 =1/(2C1 ) [C1 (NO Xk −C2 )−{C12 (C2 −NO Xk )2 −4C1 (NO Yk )AL }1/2 ]   Λi and Yk = (1/NO ) ( AL from Xk = (1/NO ) AL i=1 i=1   AL Λi )1/(AL) , where C1 = AL i=3 Zi and C2 = i=3 Zi . The Jacobian J(Λ1 , Λ2 ,. . . ,ΛAL ) = (∂Λ1/∂Xk )(∂Λ2/∂Yk )−(∂Λ1 / ∂Yk )(∂Λ2 /∂Xk ) of the transformation from (Xk , Yk , Z3 , Z4 , . . . , ZAL ) to (Λ1 , Λ2 , . . . , ΛAL ) can be obtained as

i=k

√ NO {ak (x, y) − A(γi + 1)}  +Q Pr(H1,i ) . (41) A(Aγi2 + 2γi + 1) Assume the SNRs {γi }K low. Then, we have i=1,i=k are  √ 2 Q [( NO {ak (x, y) − A(γ √ i + 1)})/ A(Aγi + 2γi + 1)] ≈ √ Q[ NO {ak (x, y) − A}/ A]. In addition, since the chi-square distribution with degree of freedom m can be approximated as a Gaussian distribution with mean m and variance 2m from the central limit theorem when m  1, we have Γ{AN √ O ) = 1−P2ANO (2NO ak (x, y))≈ √ O , NO ak (x, y)}/Γ(AN Q[ NO {ak (x, y) − A}/ A], where Pm (·) is the cdf of the chi-square distribution with degree of freedom m. In short, we have √ Γ {ANO , NO ak (x, y)} NO {ak (x, y)−A(γi + 1)}  ≈Q , Γ(ANO ) A(Aγi2 + 2γi + 1) (42) which, together with Pr(H0,i ) + Pr(H1,i ) = 1, can be used in (41) to produce 1 − FXθ (ak (x, y)|Xk = x, Ht,k ) ≈

Γ {ANO , NO ak (x, y)} Γ(ANO )

!K−1 .

J(Λ1 , Λ2 , . . . , ΛAL ) = 

. C12 (C2 −NO Xk )2 −4C1 (NO Yk )AL (45)

Then, the joint pdf fXk ,Yk (x, y|Ht,k ) of Xk and Yk can be expressed as fXk ,Yk (x, y|Ht,k )   ALNOAL+1 y AL−1 = ···  2 c1 (NO x − c2 )2 − 4c1 (NO y)AL A

· fΛ (λ|Ht,k ) dλ3 dλ4 . . . dλAL

(46)

for 0 < ALy ≤ x when AL = 3, 4, . . ., where A = {(λ3 , λ4 , . . . , λAL ): 0 < λAL ≤ λAL−1 ≤ · · · ≤ λ3 < ∞} denotes the  AL region of integration, c1 = AL i=3 λi , c2 = i=3 λi , Λ = (Λ1 , Λ2 , . . . , ΛAL ), and λ = (λ1 , λ2 , . . . , λAL ) with λ1 = Λ1|C1 =c1 ,C2 =c2 ,Xk =x,Yk =y and λ2=Λ2 |C1 =c1 ,C2 =c2 ,Xk =x,Yk =y . 2 Since (NO /PW )Δ(Rk ; L) = (1/σW )Rk RH k is an unk correlated central Wishart matrix under H0,k and a corre2 )E{Rk (n)RH lated central Wishart matrix with (1/σW k (n)} = k H 2 2 H k H k (σk /σWk ) + I A under H1,k when A = 2, 3, . . ., the joint pdfs of Λ in (44) and (46) can be obtained as [25]

(43)

A  i=1

fΛ (λ|H0,k ) = A PPENDIX D J OINT P ROBABILITY D ENSITY F UNCTION OF Xk AND Yk Let us denote by Λ1 ≥ Λ2 ≥ · · · ≥ ΛAL the ordered eigenvalues of the matrix (NO /P  W )Δ(Rk ; L). When AL = 2, we have Λ1 = (NO /2)(Xk + Xk2 − 4Yk2 ) and Λ2 = (NO /2)

ALNOAL+1 YkAL−1

A 

O −A λN i

(A − i)!(NO − i)!

i=1

·

⎧ A ⎨A−1   ⎩

(λi − λj )

i=1 j=i+1

2

⎫ ⎬ ⎭

exp −

A  i=1

λi

(47)

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

for λ1 ≥ λ2 ≥ · · · ≥ λA > 0 and A   NO −A λi (Aγk )−A+1 i=1 fΛ (λ|H1,k ) = A−2   (1 + Aγk )NO −A+1 i! i=1 % $ A−1 A   (λi − λj ) ·

i=1 j=i+1



A 



det{Gk (λ)}

(48)

(NO − i)!

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where ηk = 2ϕANO ln(ηk / {μ(θ, k)A}). Noting that Xθ can be approximated as a Gaussian random variable from the central limit theorem, the mean E(Xθ ) in μ(θ, k) for (51) can be −1 obtained as E(Xθ ) ≈ FX (0.5|H0,k ), where FXθ (x|H0,k ) = θ K 1 − i=1 Pr(Xi > x|H0,k ) [23]. In particular, since {Xi }K i=1 under H0,k can be approximated as i.i.d. Gaussian random variables with mean A and variance A/NO , we have E(Xθ ) ≈  A/NO Q−1 (0.51/K ) + A.   Now, since (ηk /2)j /Γ (A2 − 1)/2 + j + 1 is negligible when j is large, (51) can be expressed as

i=1

for λ1 ≥ λ2 ≥ · · · ≥ λA > 0, where ⎡ g1 (1+Aγk ) g2 (1+Aγk ) ⎢ g1 (1) g2 (1) ⎢ (1) ⎢ g (1) (1) g2 (1) Gk (λ) = ⎢ 1 ⎢ .. .. ⎣ . . (A−2) (A−2) (1) g2 (1) g1

PF A,k (ηk ; 1) ··· ··· ··· .. . ···

⎤ gA (1+Aγk ) gA (1) ⎥ ⎥ (1) gA (1) ⎥ ⎥ ⎥ .. ⎦ . (A−2) gA (1)

(49) (n) is an A × A matrix with gj (u) = exp(−λj /u) and gj (1) = (∂ n /∂un )gj (u)|u=1 for j = 1, 2, . . . , A. The joint pdfs of Xk and Yk can be obtained from (44) and (46) using (47) and (48). A PPENDIX E A SYMPTOTIC P ERFORMANCE A NALYSIS Approximate closed form expression of PF A,k (ηk ; 1) at low SNRs {γi }K i=1,i=k : When L = 1, since the random variable Xi is the sample mean of NO i.i.d. random vari2 2 2 O )Ri (n)2 }N ables {(1/σW n=1 and E{(1/σWi )Ri (n) } < i ∞, Xi converges almost surely to E(Xi ) as NO → ∞ [23]. In addition, when NO is large, the cdf of Xk /Yk under H0,k can be approximated as Pr((Xk /Yk ) ≤ z|H0,k ) ≈ PA2 −1 (z ) [29], where z = 2ϕANO ln(z/A) with ϕ = 1 − (2A2 + 1)/(6ANO ). Then, recollecting that E(Xk ) = A under H0,k , the false alarm probability (17) can be rewritten as PF A,k (ηk ; 1)

   Xk  ≤ ηk  θ = k, H0,k Pr(θ = k|H0,k ) ≈ 1 − Pr 4 Yk     Xk  − Pr μ(θ, k) ≤ ηk  θ = k, H0,k Pr(θ = k|H0,k ), Yk (50)

where μ(θ, k) = {E (Xθ ) + A}2 / {E (Xθ ) A} and Pr(θ = k|H0,k ) ≈ (K − 1)/K since {Xi }K i=1 under H0,k can be approximated as i.i.d. Gaussian random variables when {γi }K i=1,i=k are very low. Now, since Pr(θ = k|H0,k ) ≈ 1 for large K, we can approximate (50) as PF A,k (ηk ; 1)

   Xk  ≤ ηk  H0,k ≈ 1 − Pr μ(θ, k) Yk

≈ 1 − PA2 −1 (ηk ) 1  2j ηk   A22−1  ∞ η 2 η k − 2 k 1 2, =1−e 2 A −1 2 +j+1 j=0 Γ 2

(51)

≈1−e

η − 2k



ηk

1

 A22−1  u1

2

j=0

1 Γ

 ηk 2

A2 −1 2

2j

2, +j+1

(52)

where u1 is a positive integer that determines the accuracy of the approximation. k) is a decreasing function of E(Xθ ) ≈ Since μ(θ, A/NO Q−1 (0.51/K ) + A, when the number NO of observations increases in (51), ϕ = 1 − (2A2 + 1)/(6ANO ) increases and μ(θ, k) decreases and thus ηk increases, which implies that the false alarm probability PF A,k (ηk ; 1) ≈ 1 − PA2 −1 (ηk ) decreases as NO increases. Approximate closed form expression of ηk at low SNRs {γi }K i=1,i=k : When the target false alarm probability is αk , the threshold ηk in the k-th subband can be expressed as $ % −1 PA 2 −1 (1 − αk ) (53) ηk ≈ μ(θ, k)A exp 2ϕANO √ −1 from (51). Since PA A2 − 1)2 when A = 2 −1 (1 − αk ) ≈ (δ + 6, 7, . . . [30], the threshold ηk can be expressed as $ % √ (δ + A2 − 1)2 , (54) ηk ≈ μ(θ, k)A exp 2ϕANO where δ=

−d2 −



d22 − 4d1 d3 2d1

(55)

with d1 = −0.6763, d2 = −1.2451, and d3 = − ln(2αk ) for αk < 0.5. A PPENDIX F P ROBABILITY T HAT THE S UBBAND W ITH THE L OWEST PSD IS U NDER THE N ULL H YPOTHESIS Noting that the random variables {Xi }K i=1 are independent of each other, the probability ζ[1] that the subband with the lowest PSD is under the null hypothesis can be obtained as    Pr Xj is X[1]  H0,j Pr(H0,j )   ζ[1] = Pr Xj is X[1] =

Pr(H0,j )   Pr Xj is X[1]

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER 2014

∞ ·

    Pr {Xm > x}K m=1,m=j Xj = x, H0,j

[14]

−∞

· fXj (x|H0,j ) dx =

Pr(H0,j )   Pr Xj is X[1]

[15]

∞  K 8 −∞

F0,i (x) Pr(H0,i )

[16]

i=1 i=j

9 + F1,i (x) Pr(H1,i ) fXj (x|H0,j ) dx,

[17]

(56)

where Ft,i (x) = 1 − FXi (x|Ht,i ) for t = 0 and 1. When Pr(H0,j ) = p and γi = γj for all i and j, since Pr(Xj is X[1] ) = (1/K), (56) can be expressed as ∞

[18] [19] [20]

{1 − pFXi (x|H0,i )

ζ[1] = pK −∞

− (1 − p)FXi (x|H1,i )}K−1 fXi (x|H0,i ) dx.

[21]

(57) [22]

ACKNOWLEDGMENT The authors would like to thank the Editor and three anonymous reviewers for their constructive suggestions and helpful comments. R EFERENCES [1] T. Yücek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” IEEE Commun. Surveys Tuts., vol. 11, no. 1, pp. 116–130, 2009. [2] J. Lunden, S. A. Kassam, and V. Koivunen, “Robust nonparametric cyclic correlation-based spectrum sensing for cognitive radio,” IEEE Trans. Signal Process., vol. 58, no. 1, pp. 38–52, Jan. 2010. [3] H. G. Kang, I. Song, S. Yoon, and Y. H. Kim, “A class of spectrumsensing schemes for cognitive radio under impulsive noise circumstances: Structure and performance in nonfading and fading environments,” IEEE Trans. Veh. Technol., vol. 59, no. 9, pp. 4322–4339, Nov. 2010. [4] E. Axell, G. Leus, E. G. Larsson, and H. V. Poor, “Spectrum sensing for cognitive radio: State-of-the-art and recent advances,” IEEE Signal Process. Mag., vol. 29, no. 3, pp. 101–116, May 2012. [5] F. F. Digham, M.-S. Alouini, and M. K. Simon, “On the energy detection of unknown signals over fading channels,” in Proc. IEEE Int. Conf. Commun., Anchorage, AK, USA, May 2003, vol. 5, pp. 3575–3579. [6] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008. [7] A. Mariani, A. Giorgetti, and M. Chiani, “Effects of noise power estimation on energy detection for cognitive radio applications,” IEEE Trans. Commun., vol. 59, no. 12, pp. 3410–3420, Dec. 2011. [8] Y. Zeng and Y.-C. Liang, “Spectrum-sensing algorithms for cognitive radio based on statistical covariances,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp. 1804–1815, May 2009. [9] Y. Zeng and Y.-C. Liang, “Eigenvalue-based spectrum sensing algorithms for cognitive radio,” IEEE Trans. Comm., vol. 57, no. 6, pp. 1784–1793, June 2009. [10] R. Zhang, T. J. Lim, Y.-C. Liang, and Y. Zeng, “Multi-antenna based spectrum sensing for cognitive radios: A GLRT approach,” IEEE Trans. Commun., vol. 58, no. 1, pp. 84–88, Jan. 2010. [11] A. Taherpour, M. Nasiri-Kenari, and S. Gazor, “Multiple antenna spectrum sensing in cognitive radios,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 814–823, Feb. 2010. [12] P. Wang, J. Fang, N. Han, and H. Li, “Multiantenna-assisted spectrum sensing for cognitive radio,” IEEE Trans. Veh. Technol., vol. 59, no. 4, pp. 1791–1800, May 2010. [13] D. Ramirez, G. Vazquez-Vilar, R. Lopez-Valcarce, J. Via, and I. Santamaria, “Detection of rank-P signals in cognitive radio net-

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Taehun An (S’07) received the B.S. degree in electronics engineering from Sungkyunkwan University, Suwon, Korea, in 2006 and the M.S.E. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2008. He is currently working toward the Ph.D. degree at KAIST. Since March 2006, he has been a Teaching and Research Assistant in the Department of Electrical Engineering, KAIST. His research interests include mobile communications, multiple input multiple output systems, detection and estimation theory, and statistical signal processing.

AN et al.: DETECTION OF SIGNALS WITH OBSERVATIONS IN MULTIPLE SUBBANDS

Iickho Song (S’80–M’87–SM’96–F’09) received the B.S.E. (magna cum laude) and M.S.E. degrees in electronics engineering from Seoul National University, Seoul, Korea, in 1982 and 1984, respectively, and the M.S.E. and Ph.D. degrees in electrical engineering from the University of Pennsylvania, Philadelphia, PA, USA, in 1985 and 1987, respectively. He was a Member of the Technical Staff at Bell Communications Research in 1987. In 1988, he joined the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, where he is currently a Professor. Prof. Song served as the Treasurer of the IEEE Korea Section in 1989, as an Editor of the Journal of the Acoustical Society of Korea (ASK), and as an Editor of the Journal of the Institute of Electronics Engineers of Korea (IEEK). He has also served as an Editor of the Journal of the Korean Institute of Communications and Information Sciences (KICS) since 1995, and as an Editor of the Journal of Communications and Networks since 1998. He has coauthored Advanced Theory of Signal Detection (Springer, 2002), Random Processes (in Korean; Saengneung, 2004), Signals and Systems (Hongreung, 2008; Springer, 2009), Principles of Random Processes (in Korean; Kyobo, 2009), and Random Variables and Random Processes (in Korean; Freedom Academy, 2014), and has published a number of papers on signal detection and mobile communications. His research interests include detection and estimation theory, statistical communication theory and signal processing, and wireless communications. Dr. Song is a Fellow of Korean Academy of Science and Technology (KAST). He is also a Member of the ASK, IEEK, KICS, and Korea Institute of Information, Electronics, and Communication Technology, and a Fellow of the Institution of Engineering and Technology (IET). Prof. Song has received many awards including the Young Scientists Award (KAST, 2000), Achievement Award (IET, 2006), and Hae Dong Information and Communications Academic Award (KICS, 2006).

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Seungwon Lee (S’14) received the B.S.E. degree in electronics engineering from Kyung Hee University, Yongin, Korea, in 2010, and the M.S.E. degree in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2012. He is currently working toward the Ph.D. degree at KAIST. Since February 2010, he has been a Teaching and Research Assistant in the Department of Electrical Engineering, KAIST. His research interests include mobile communications, detection and estimation theory, and statistical signal processing.

Hwang-Ki Min received the B.S.E. and M.S.E. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree at KAIST. His research interests include pattern recognition, machine learning theory, and statistical signal processing.